In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series. 

If the function $f: \mathbb{C} \to \mathbb{C} $ satisfies certain  conditions$\raise0.9ex\small \hbox{a}$, sums can be evaluated by means of this theorem according to the following result: $$\lim_{N \to +\infty} \sum_{k=-N}^{N} f(k) = - \ \{ \text{sum of the residues of } \pi f(z) \cot(\pi z) \text{ at the poles of }f(z) \} . \tag{*}\label{star}$$

It appears to be the case that residues also exist and can be computed in quaternionic analysis: 

 - According to <a href="https://arxiv.org/abs/math/0209166v1">Differentiable functions of quaternion variables</a> by Lüdkovsky and Oystaeyen (see also the MSE question <a href="https://math.stackexchange.com/questions/1480395/is-there-a-residue-theorem-for-quaternions">Is there a residue theorem for Quaternions?</a>), one can obtain residues of functions in quaternionic analysis as well. 
 - In Section 9 of <a href="https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-analysis-memo.pdf">Quaternionic analysis</a>, author Anthony Sudbery also shows there's an analogue of the residue theorem in quaternionic analysis.

I am curious$\raise0.9ex\small\hbox{b}$ as to whether it may be possible to formulate an analogue of \eqref{star} for a function $g$ of a quaternionic variable. 

**Questions**: 

 1. Does an analogue of \eqref{star} exist for functions $g$ of a quaternionic variable? So functions like $g: \mathbb{H} \to \mathbb{H}$? 
 2. If so, what conditions must $g$ satisfy? 
 3. Can this quaternionic analogue of the residue formula for series be applied to evaluate infinite series that cannot be evaluated by means of the residue sum formula from complex analysis? 

*Notes*: 

$\raise0.9ex\small\hbox{a}$ The complex function must satisfy the condition that $$\lvert f(z)\rvert < \frac{M}{z^{k}} $$ — where $k>1$ and $M$ are constants independent of $N$ — along the path $C_{N}$. This path goes counterclockwise along the rectangle $(N + (1/2))(1-i)$, $(N+(1/2))(1+i)$, $(N+(1/2))(-1-i)$ and $(N+(1/2))(-1-i)$ and it encloses the poles of $f$. See pp. 3–5 of <a href="http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf">Hughes - Infinite Series and the Residue Theorem</a> for more details. 

$\raise0.9ex\small\hbox{b}$ I've asked a similar question <a href="https://math.stackexchange.com/questions/1480395/is-there-a-residue-theorem-for-quaternions">Is there a residue theorem for Quaternions?</a> on MSE.